Quantum physics is a fascinating area from a computational viewpoint. The features that make quantum systems prohibitively hard to simulate classically are precisely the aspects exploited by quantum computation to obtain exponential speedups over classical computers. In this talk I will survey our current understanding of the power (and limits) of quantum computers, and prospects for experimentally realizing them in the near future. I will also touch upon insights from quantum comuptation that have resulted in new classical algorithms for efficient simulation of certain important quantum systems.